On the Universal Encoding Optimality of Primes

The factorial-additive optimality of primes, i.e., that the sum of prime factors is always minimum, implies that prime numbers are a solution to an integer linear programming (ILP) encoding optimization problem. The summative optimality of primes follows from Goldbach’s conjecture, and is viewed as an upper efficiency limit for encoding any integer with the fewest possible additions. A consequence of the above is that primes optimally encode—multiplicatively and additively—all integers. Thus, the set P of primes is the unique, irreducible subset of ℤ—in cardinality and values—that optimally encodes all numbers in ℤ, in a factorial and summative sense. Based on these dual irreducibility/optimality properties of P, we conclude that primes are characterized by a universal “quantum type” encoding optimality that also extends to non-integers.